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well-defined

A mathematical concept is well-defined (German wohldefiniert, French bien défini), if its contents is
independent on the form or the alternative representative which is used for defining it.

For example, in defining the powerxrsuperscriptxrx^{r} with xxx a positivereal and rrr a rational number,
we can freely choose the fraction form mnmn\frac{m}{n} (m∈ℤmℤm\in\mathbb{Z}, n∈ℤ+nsubscriptℤn\in\mathbb{Z}_{+}) of rrr and take

xr:=xmnassignsuperscriptxrnsuperscriptxmx^{r}\;:=\;\sqrt[n]{x^{m}}

and be sure that the value of xrsuperscriptxrx^{r} does not depend on that choice (this is justified in the entry fraction power). So,
the xrsuperscriptxrx^{r} is well-defined.

In many instances well-defined is a synonym for the formal definition of a function between sets. For example,
the function f⁢(x):=x2assignfxsuperscriptx2f(x):=x^{2} is a well-defined function from the real numbers to the real numbers because
every input, xxx, is assigned to precisely one output, x2superscriptx2x^{2}. However, f⁢(x):=±xassignfxplus-or-minusxf(x):=\pm\sqrt{x} is not well-defined
in that one input xxx can be assigned any one of two possible outputs, xx\sqrt{x} or -xx-\sqrt{x}.

Certainly every input has an output, for instance, f⁢(1/2)=3f123f(1/2)=3. However, the expression is not
well-defined since 1/2=2/412241/2=2/4 yet f⁢(1/2)=3f123f(1/2)=3 while f⁢(2/4)=6f246f(2/4)=6 and 3≠6363\neq 6.

One must question whether a function is well-defined whenever it is defined on a domain of equivalence classes
in such a manner that each output is determined for a representative of each equivalence class. For example, the
function f⁢(a/b):=a+bassignfababf(a/b):=a\!+\!b was defined using the representative a/baba/b of the equivalence class of fractions
equivalent to a/baba/b.